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We develop a model of gamma-ray flares of the Crab Nebula resulting from the magnetic reconnection events in a highly magnetised relativistic plasma. We first discuss physical parameters of the Crab Nebula and review the theory of pulsar winds and termination shocks. We also review the principle points of particle acceleration in explosive reconnection events [Lyutikov et al., J. Plasma Phys., vol. 83(6), p. 635830601 (2017a); J. Plasma Phys., vol. 83(6), p. 635830602 (2017b)]. It is required that particles producing flares are accelerated in highly magnetised regions of the nebula. Flares originate from the poleward regions at the base of the Crab’s polar outflow, where both the magnetisation and the magnetic field strength are sufficiently high. The post-termination shock flow develops macroscopic (not related to the plasma properties on the skin-depth scale) kink-type instabilities. The resulting large-scale magnetic stresses drive explosive reconnection events on the light-crossing time of the reconnection region. Flares are produced at the initial stage of the current sheet development, during the X-point collapse. The model has all the ingredients needed for Crab flares: natural formation of highly magnetised regions, explosive dynamics on the light travel time, development of high electric fields on macroscopic scales and acceleration of particles to energies well exceeding the average magnetic energy per particle.

The extreme properties of the gamma-ray flares in the Crab nebula present a clear challenge to our ideas on the nature of particle acceleration in relativistic astrophysical plasma. It seems highly unlikely that standard mechanisms of stochastic type are at work here and hence the attention of theorists has switched to linear acceleration in magnetic reconnection events. In this series of papers, we attempt to develop a theory of explosive magnetic reconnection in highly magnetized relativistic plasma which can explain the extreme parameters of the Crab flares. In the first paper, we focus on the properties of the X-point collapse. Using analytical and numerical methods (fluid and particle-in-cell simulations) we extend Syrovatsky’s classical model of such collapse to the relativistic regime. We find that the collapse can lead to the reconnection rate approaching the speed of light on macroscopic scales. During the collapse, the plasma particles are accelerated by charge-starved electric fields, which can reach (and even exceed) values of the local magnetic field. The explosive stage of reconnection produces non-thermal power-law tails with slopes that depend on the average magnetization
$\unicode[STIX]{x1D70E}$
. For sufficiently high magnetizations and vanishing guide field, the non-thermal particle spectrum consists of two components: a low-energy population with soft spectrum that dominates the number census; and a high-energy population with hard spectrum that possesses all the properties needed to explain the Crab flares.

Using analytical and numerical methods (fluid and particle-in-cell simulations) we study a number of model problems involving merger of magnetic flux tubes in relativistic magnetically dominated plasma. Mergers of current-carrying flux tubes (exemplified by the two-dimensional ‘ABC’ structures) and zero-total-current magnetic flux tubes are considered. In all cases regimes of spontaneous and driven evolution are investigated. We identify two stages of particle acceleration during flux mergers: (i) fast explosive prompt X-point collapse and (ii) ensuing island merger. The fastest acceleration occurs during the initial catastrophic X-point collapse, with the reconnection electric field of the order of the magnetic field. During the X-point collapse, particles are accelerated by charge-starved electric fields, which can reach (and even exceed) values of the local magnetic field. The explosive stage of reconnection produces non-thermal power-law tails with slopes that depend on the average magnetization
$\unicode[STIX]{x1D70E}$
. For plasma magnetization
$\unicode[STIX]{x1D70E}\leqslant 10^{2}$
the spectrum power-law index is
$p>2$
; in this case the maximal energy depends linearly on the size of the reconnecting islands. For higher magnetization,
$\unicode[STIX]{x1D70E}\geqslant 10^{2}$
, the spectra are hard,
$p<2$
, yet the maximal energy
$\unicode[STIX]{x1D6FE}_{\text{max}}$
can still exceed the average magnetic energy per particle,
${\sim}\unicode[STIX]{x1D70E}$
, by orders of magnitude (if
$p$
is not too close to unity). The X-point collapse stage is followed by magnetic island merger that dissipates a large fraction of the initial magnetic energy in a regime of forced magnetic reconnection, further accelerating the particles, but proceeds at a slower reconnection rate.

We show that under conditions of strong density modulation the effects of magnetospheric scintillations in diffractive and refractive regimes may be observable. The most distinctive feature of the magnetospheric scintillations is their independence on frequency. Results based on diffractive scattering due to small scale in homogeneities give a scattering angle that may be as large as 0.1 radians, and a typical decorrelation time of 10−8 seconds. Refractive scattering due to large scale inhomogeneities is also possible, with a typical angle of 10−3 radians and a correlation time of the order of 10−4 seconds. Temporal variation in the plasma density may also result in a delay time of the order of 10−4 seconds. The different scaling of the above quantities with frequency may allow one to distinguish the effects of propagation through a pulsar magnetosphere from the interstellar medium. In particular, we expect that the magnetospheric scintillations are relatively more important for nearby pulsars when observed at high frequencies.

Relativistic plasma masers operating on the anomalous cyclotron-Cherenkov resonance ω − k||υ|| + ωB/ϒres = 0 and the Cherenkov-drift resonance ω − k||υ|| − kx/ud = 0, are capable of explaining the main observational characteristics of pulsar radio emission. Both electromagnetic instabilities are due to the interaction of the fast particles from the primary beam and from the tail of the secondary pairs distribution with the normal modes of a strongly magnetized one-dimensional electron-positron plasma. In a typical pulsar both resonances occur in the outer parts of magnetosphere at rres ≈ 109cm.

Diffractive and refractive magnetospheric scintillations may allow direct probing of the plasma inside the pulsar light cylinder. The unusual electrodynamics of the strongly magnetized electron-positron plasma allows separation of the magnetospheric and interstellar scattering. The most distinctive feature of the magnetospheric scintillations is their independence of frequency. Diffractive scattering due to small scale inhomogeneities produces a scattering angle that may be as large as 0.1 radians, and a typical decorrelation time of 10−8 seconds. Refractive scattering due to large scale inhomogeneities is also possible, with a typical angle of 10−3 radians and a correlation time of the order of 10−4 seconds. Some of the magnetospheric propagation effects may have already been observed.

Beam instabilities in the strongly magnetized electron–positron plasma
of a pulsar magnetosphere are considered. We analyse the resonance conditions and
estimate the growth rates of the Cherenkov and cyclotron instabilities of the ordinary
(O), extraordinary (X) and Alfvén modes in two limiting regimes: kinetic and
hydrodynamic. The importance of the different instabilities as a source of coherent
pulsar radiation generation is then estimated, taking into account the angular
dependence of the growth rates and the limitations on the length of the coherent
wave–particle interaction imposed by the curvature of the magnetic field lines. We
conclude that in the pulsar magnetosphere, Cherenkov-type instabilities occur in
the hydrodynamic regime, while cyclotron-type instabilities occur in the kinetic
regime. We argue that electromagnetic cyclotron-type instabilities on the X, O and
probably Alfvén waves are more likely to develop in the pulsar magnetosphere.

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